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The Physical Role of Gravitational and Gauge Degrees of Freedom in General Relativity - II: Dirac versus Bergmann observables and the Objectivity of Space-Time
(abridged)The achievements of the present work include: a) A clarification of
the multiple definition given by Bergmann of the concept of {\it (Bergmann)
observable. This clarification leads to the proposal of a {\it main conjecture}
asserting the existence of i) special Dirac's observables which are also
Bergmann's observables, ii) gauge variables that are coordinate independent
(namely they behave like the tetradic scalar fields of the Newman-Penrose
formalism). b) The analysis of the so-called {\it Hole} phenomenology in strict
connection with the Hamiltonian treatment of the initial value problem in
metric gravity for the class of Christoudoulou -Klainermann space-times, in
which the temporal evolution is ruled by the {\it weak} ADM energy. It is
crucial the re-interpretation of {\it active} diffeomorphisms as {\it passive
and metric-dependent} dynamical symmetries of Einstein's equations, a
re-interpretation which enables to disclose their (nearly unknown) connection
to gauge transformations on-shell; this is expounded in the first paper
(gr-qc/0403081). The use of the Bergmann-Komar {\it intrinsic
pseudo-coordinates} allows to construct a {\it physical atlas} of 4-coordinate
systems for the 4-dimensional {\it mathematical} manifold, in terms of the
highly non-local degrees of freedom of the gravitational field (its four
independent {\it Dirac observables}), and to realize the {\it physical
individuation} of the points of space-time as {\it point-events} as a
gauge-fixing problem, also associating a non-commutative structure to each
4-coordinate system.Comment: 41 pages, Revtex
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